Geodesic
Groups of automatic automorphisms of some automatic structures
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 3 (2006), pp. 145-152.

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We prove that the group $\operatorname{Aut}_a(\{0,1\}^\ast)$ of all automatic automorphisms of the regular set $\{0,1\}^\ast$ and the group $\operatorname{Aut}_a(\mathbb{Q})$ of all automatic automorphisms of the automatic model $\mathbb{Q}=(\{0,1\}^\ast,\preccurlyeq_{lex})$ have undecidable theories, which implies that they have no automatic presentations. 
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N. S. Vinokurov. Groups of automatic automorphisms of some automatic structures. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 3 (2006), pp. 145-152. http://geodesic.mathdoc.fr/item/SEMR_2006_3_a7/

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